Polynomial 6j–symbols and states sums

Geer, Nathan; Patureau-Mirand, Bertrand

doi:10.2140/agt.2011.11.1821

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Polynomial $6j$–symbols and states sums

Nathan Geer and Bertrand Patureau-Mirand

Algebraic & Geometric Topology 11 (2011) 1821–1860

DOI: 10.2140/agt.2011.11.1821

Abstract

For a given $2 r$ –th root of unity $ξ$ , we give explicit formulas of a family of $3$ –variable Laurent polynomials $J_{i, j, k}$ with coefficients in $ℤ [ξ]$ that encode the $6 j$ –symbols associated with nilpotent representations of $U_{ξ} (s l (2))$ . For a given abelian group $G$ , we use them to produce a state sum invariant $τ^{r} (M, L, h_{1}, h_{2})$ of a quadruplet (compact $3$ –manifold $M$ , link $L$ inside $M$ , homology class $h_{1} \in H_{1} (M, ℤ)$ , homology class $h_{2} \in H_{2} (M, G)$ ) with values in a ring $R$ related to $G$ . The formulas are established by a “skein” calculus as an application of the theory of modified dimensions introduced by the authors and Turaev in [Compos. Math. 145 (2009) 196–212]. For an oriented $3$ –manifold $M$ , the invariants are related to $τ (M, L, ϕ \in H^{1} (M, ℂ^{*}))$ defined by the authors and Turaev in [arXiv:0910.1624] from the category of nilpotent representations of $U_{ξ} (s l (2))$ . They refine them as $τ (M, L, ϕ) = \sum_{h_{1}} τ^{r} (M, L, h_{1}, \tilde{ϕ})$ where $\tilde{ϕ}$ correspond to $ϕ$ with the isomorphism $H_{2} (M, ℂ^{*}) ≃ H^{1} (M, ℂ^{*})$ .

Keywords

$6j$–symbols, state sum, skein calculus, quantum groups, $3$–manifolds

Mathematical Subject Classification 2000

Primary: 57M27, 81Q99

References

Publication

Received: 13 July 2010

Accepted: 8 October 2010

Published: 12 June 2011

Authors

Nathan Geer
Mathematics & Statistics Utah State University 3900 Old Main Hill Logan, UT 84322 USA

Bertrand Patureau-Mirand
LMAM Universite de Bretagne-Sud BP 573 F-56017 Vannes France

Volume 11, issue 3 (2011)